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TENSOR PRODUCT OF A NEAR-FIELD SPACE AND SUB NEAR-FIELD SPACE OVER A NEAR-FIELD
Dr. N. V. NAGENDRAM*
Professor of Mathematics,
Kakinada Institute of Technology & Science (K.I.T.S.)
Department of Humanities & Science (Mathematics), Tirupathi (vill) Peddapuram (M), Divili
533 433, East Godavari District. Andhra Pradesh. INDIA.
Received On: 21-05-17; Revised & Accepted On: 03-06-17)
ABSTRACT
A
tensor product of near-field space and sub near-field space over a near-field defined. But it turned out to be an abelian group. In order to avoid this special situation, this concept is generalized further by Dr N V Nagendram in this paper. Now there are two tensor products for the same pair of sub field spaces of a field space over a near-field. This situation fits better in the theory of near-field spaces and their sub near-field spaces over a near-near-field. It has been seen here that there may be two dual near-field spaces for a pair of sub near-field spaces over a near-field.Keywords: sub near-field space, near-field space, semi simple near-field space, ordered Euclidean near-field spaces,
vector lattices.
2000 Mathematics Subject Classification: 43A10, 46B28, 46H25, 46H99, 46L10, 46M20, 51 M 10, 51 F 15, 03 B 30.
SECTION 1: INTRODUCTION
We write maps on the right and hence use left near-field spaces and the traditional sub near-field spaces are right sub near-field spaces.
Definition 1.1 (N-sub near-field space): Let (N, +, .) be a left near-field space. A sub near-field space (M, +) is called an N-sub near-field space i.e. traditional one if there is a near-field space homo-morphism θ : N → Map(M). As usual, we write gn to mean g(nθ) for g ∈ M and n∈ N. In this case the group elements distribute over the near-field spaces.
Definition 1.2 (Complementary N-near-field space): M is called a complementary N-sub near-field space or N – co sub near-field space, for short, if there is a semi sub near-field space elements distribute over the sub near-field space elements and the action of N is usually written on the left of the elements of M.
Definition 1.3 ((N, T) – bi sub near-field space): Let N and T be two left near-field spaces. A sub near-field space M is called an (N, T) – bi sub near-field space if
(a) M is an N-co sub near field space
(b) M is an T-sub near-field space and (c) (ng)t = n(gt),
∀
g∈
M, n∈
N, t∈
T.Definition 1.4 (left strong N-sub near-field space): M is called left strong N-sub near-field space if the action of N is defined on the left of M satisfying the following conditions ∀n, n′∈ N and g, g’ ∈ M
(a) (nn’)g = n(n’g)
(b) n (g + g’ ) = ng + ng’ and (c) (n + n’)g = ng + n’g.
Note-1.5: A right strong N-sub near-field space is defined similarly. (N, +) is an (N – N ) – bi sub near-field space for the left as well as right near-field space N over a near-field. If N is distributive near-field space then (N, +) is a left as well as right strong N-sub near-field space. Many more examples of these structures are given in near-field space related topic.
Corresponding Author: Dr. N. V. Nagendram, Professor of Mathematics, Kakinada Institute of
Technology & Science, Tirupathi (v), Peddapuram(M), Divili 533 433, East Godavari District,
Definition 1.6 (N-homomorphism): Let M and K be two N-sub near-field spaces (N-co sub near-field space. A sub near-field space homomorphism θ : M → K is called an N-homomorphism if for any g∈ M and n∈ N, (gn)θ = (gθ)n, ((rg)θ = r(gθ)).
Note-1.7: An (N – T) – homomorphism for (N – T)-bi sub near-field space are defined in a similar way.
SECTION 2: TENSOR PRODUCT
Let N be left near-field space, A an N-sub near-field space and B an N-co sub near-field space. Let E be the free collection of sub near-field spaces on A × B. Let P and Q be the normal sub near-field spaces of E generated by {(a + a|, b) – (a′, b) – (a, b), (ar, b) – (a, rb) | a, a′∈ A, b∈ B, r∈ N} and
{(a, b + b′) – (a, b′) – (a, b), (ar, b) – (a, rb) | a∈ A, b, b′∈ B, r∈ N} respectively.
We call E/P the left tensor product of A and B and denote it by AN ⊗ B and call E/Q the right tensor product of A and B
and denote it by A ⊗NB. The coset (a, b) + P is denoted by al ⊗ b and (a, b) + Q is denoted by a ⊗rb. The coser P is
denoted by 0 in both cases. Since E is generated by A × B, E/P = AN ⊗ B and E/Q = A ⊗ NB are generated by
{al ⊗ b/a ∈ A, b ∈ B} and {a ⊗rb|a ∈ A, b ∈ B} respectively. An element of AN⊗ B (A ⊗NB) is a finite sum of the
form
∑
∈
i(
a
il
⊗
⊗
b
i)(
∑
∈
i(
a
i⊗
rb
i))
, where each ∈i = ± 1.The following result is a direct consequence of the definition of tensor product of near-field space and sub near-field space over a near-field.
Theorem 2.1: ∀a, a′∈ A, b ∈ B, r ∈ N the following are satisfied in AN⊗B:
(i) (a + a
′
)l⊗
b = al⊗
b + a′
l⊗
b(ii) arl ⊗b = a;
⊗
rb(iii) 0Al
⊗
b = 0(iv) (-a)l
⊗
b = - ( al⊗
b ).Theorem 2.2: ∀a∈ A, b, b′∈ B, r ∈ N the following are satisfied in A⊗ NB:
(i) (a )l
⊗
(b + b′
)′
= a⊗
rb + a⊗
rb′
(ii) ar
⊗
rb = a⊗
rrb(iii) a
⊗
rBb = 0(iv) a
⊗
r( -b) = - ( a⊗
rb ).Remark 2.3: In general ( a + a′ )r ≠ ar + a′r in A as A is an N-sub near-field space. But we have (a + a′)rl⊗ b = (a + a′)l⊗ rb = al⊗ rb + a′l⊗ rb
= arl⊗ b + a′rl⊗ b
= ( ar + a′r )l⊗ b.
This shows that in AN ⊗ B with b ≠ 0.
Remark 2.4: It is possible that a ⊗ rb = 0 in A⊗ NB, with b ≠ 0. Later on we will show these by examples and we
generalize the definition of a middle linear map.
Definition 2.5 (left N-middle linear map(LNMLM)): Let N, A and B be as before and let C be any sub near-field space with a map f : A × B → C. then we call f a left N-middle linear map if (a + a′, b) f = (a, b ) f + (a′ , b) f, (ar, b) f = (a, rb ) f , ∀a, a′∈ A, b ∈ B, r ∈ N.
Definition 2.6 (Right N-middle linear map(RNMLM)): Let N , A and B be as before and let C be any sub near-field space with a map f : A × B → C. then we call f a left N-middle linear map if (a, b + b′) f = (a, b ) f + (a , b′) f, (ar, b) f = (a, rb ) f , ∀a, a′∈ A, b,b′∈ B, r ∈ N.
It is easy to see that θl = jπP : A × B → AN⊗ B = E/P, (a, b)
al⊗ b andθl = jπQ : A × B → A ⊗NB = E/Q, (a, b)
a ⊗ rb areLNMLM and RNMLM respectively. Here j : A × B → E is the inclusion map and πP : E → E/P and πQ : E → E/Q are
the natural homomorphisms. We call θl (θr) the canonical LNMLM (RNMLM).
Theorem 2.7: Let N be a left-near field space, A be an N-sub near-field space and B an N-co sub near-field space. Let C be a group with a function f : A × B → C, and θl (θr ) be as above.
(a). If f is a LNMLM, then there exists a unique sub near-field space homomorphism g : AN⊗ B → C, such that θlg = f
(b). If f is a RNMLM, then there exists a unique sub near-field space homomorphism g : A ⊗NB → C, such that
θrg = f .
Proof: Given N be a left-near field space, A be an N-sub near-field space and B an N-co sub near-field space. Let C be a group with a function f : A × B → C.
To prove (a): Let us consider the following diagrammatic expression of near-field spaces over a near-field N.
where h is the unique homomorphism extending f, as E is the free sub near-field space on A × B. Since f is an LNMLM, P ⊆ Ker h. This gives us a unique sub near-field space homomorphism g: E/P → C, such that πPg = h. It
follows then θlg = jπLg = jh = f. Now for the uniqueness let g′be another homomorphism from E/P to C with θlg′ = f.
then
(al⊗ b)g′ = (a, b) θlg′ = (a, b) f = (a, b)jh = (a, b) jπLg = (a, b)θlg = (al⊗ b)g.
Therefore, g and g′ agree on the generators of AN ⊗ B and hence are equal.
Hence proved (b). In similar manner (b) can be proved. This comples the proof of the theorem.
Corollary 2.8: Let A, A′ be N-sub near-field spaces and B, B′ be N-co sub near-field spaces over a near-field N, f : A → A′ and g : B → B′ be N-homomorphisms of N-sub near-field spaces and N-co sub near-field spaces
respectively. Then there are unique sub near-field space homomorphisms φ : AN ⊗ B → A′ N ⊗ B′ and
Ψ : A ⊗N B → A′⊗N B′ such that (al⊗ b)φ = af l⊗ bg and ( a ⊗rb) Ψ = af⊗r bg.
Note-2.9: The homomorphism φ and Ψ in the above corollary are denoted by f l⊗g and f ⊗rg respectively.
If A, A′, A′′ are N-sub near-field spaces of a near-field space over a near-field N and B, B′, B′′ are N-sub near-field spaces of a near-field space over a near-field N with N-homomorphisms f : A → A′ , f ′ : A → A ′′ , g : B → B′ and g ′ : B → B′′ , then ( f l⊗g ) ( f′l⊗g′ ) = f f ′ l⊗gg ′ and ( f ⊗rg ) ( f′⊗⊗rg′) = f f ′⊗rgg ′. Moreover, if f and g
are isomorphisms then fl⊗g and f ⊗rg are isomorphisms.
Theorem 2.10: Let N and T be left near-field spaces over a near-field, A an (N – T) - bi sub near-field space and B an T- co sub near-field space over a near-field. The A T⊗ B and A ⊗T B are N – co sub near-field spaces over a near-field
N.
Proof: ∀ n ∈ N, define αn : A × B → A T⊗ B by (a, b) αn = na l⊗ b, for all (a, b) ∈ A × B. We claim that αn is a
LTMPM. For all a , a′∈ A, b ∈ B and t ∈T then we have the following relation as
(a + a′, b) αn = n (a + a′) l⊗ b = (na + na′ ) l⊗ b = na l⊗ b + na′ l⊗ b = (a, b) αn + (a′, b) αn.
(at, b) αn = n(at) l⊗ b = (na) t l⊗ b = na l⊗ sb = (a, sb) αn.
By theorem 2.9 there is a unique endomorphism βn of AT ⊗ B such that θl βn = αn, where θl is the canonical
LTMPM : A × B → A T⊗ B. The action of N on A T⊗ B is now defined by nu = uβn for n ∈ N and u ∈ A T ⊗ B. We
claim that this action defines AT ⊗ B as an N-co sub near-field space over a near-field N. For all u, u′∈ A T⊗ B and
n, n′∈ N we have n( u + u′ ) = ( u + u′ ) βn = u βn + u′βn = nu +nu′. In order to prove that (nn′)u = n(n′u) , it is enough
to prove that βnn′ = βn′βn for all n, n′∈ N. we look at their action on generators of AT ⊗ B.
(a l⊗ b) βnn′= (a, b)θlβnn′= (a, b) αnn′ = (nn′)a l⊗ b = n(n′a) l⊗ b = (n′a, b)αn = (n′a l⊗ b) βn = (a, b) αn′ βn = (a l ⊗ b)
βn′βn. The fact that A T ⊗ B is N-co sub near-field space over a near-field and similar manner one can prove that
Note-2.11: NN⊗ B and N ⊗N B are co sub near-field spaces for any N-co sub near-field space B over a near-field.
Note-2.12: Let N be a left near-field space with 1, and B be any unital N-co sub near-field space. Then for n ∈ N and b ∈ B, we have n l ⊗ b = 1 l⊗ nb and n ⊗n b = 1 ⊗n nb. Therefore we have the following results:
(a) N N⊗ B is generated by { 1 l⊗ b | b ∈ B }
(b) N ⊗N B is generated by {0 ⊗n b, 1 ⊗n b / b ∈ B \ {0}}.
(c) θb : N → N N⊗ B defined by nθb = n l⊗ b is a sub near-field space homomorphism .
(d) θb : N → N ⊗N B defined by nθb = n ⊗nb is not a sub near-field space homomorphism in general.
Theorem 2.13: let N and T be left near-field spaces, A an N-sub near-field space and B an (N – T)-bi sub near-field space. Then A N ⊗ B is an T-sub near-field space with T acting on the right. If in addition B is a right strong T-sub
near-field space then A ⊗NB is also a T-co sub near-field space with T acting on the right.
Proof: For t ∈ T define αt : A × B → AN⊗ B by (a, b) αt = al⊗ bt. It is easy to see that αt is a LNMLM.
This gives us a unique endomorphism βt of AN⊗ B such that θlβt = αt, where θl: A × B → AN ⊗ B is the canonical
LNMLM. For all ( al ⊗ b ) ∈ AN⊗ B, we have (al⊗ b ) βt = (a, b) θlβt = (a, b) αt = a l⊗ (bt).
Now we define an action of T on AN⊗ B by ut = uβt∀ u ∈ AN ⊗ B. Clearly, (u + u′)t = ut + u′t, ∀ u , u′∈ AN⊗ B.
For the other condition we need to show that βtt′ = βtβt′∀ t, t′ ∈ T. It is enough to look at their behaviour on the
generators.
(a l⊗ b) βtt′ = a l⊗ b (tt′) = a l⊗ (bt) t′ = (a l⊗ (bt)) βt′ = (a l⊗ b) βtβt′. The second part can be proved in similar
manner. This completes the proof of the theorem.
Corollary 2.14: AN⊗ B is an N-co sub near-field space with N acting on the right. If N is distributive near-field space
then AN⊗ N and A ⊗NN is an N- co sub near-field space with N acting on the right.
Note-2.15: Let N be a left near-field space with 1 and A be a unital N-sub near-field space. Then we have AN⊗ N is
generated by notation of a set {a1⊗ 1, a l⊗ 0 | a ∈ A \ {0}} and A ⊗NN is generated by {a ⊗r1 | a ∈ A}.
Note-2.16: If A is any sub near-field space and B is an abelian sub near-field space then AZ⊗ B and A ⊗ZB are right
Z-co sub near-field spaces and hence are abelian sub near-field spaces over a near-field N.
Note-2.17: If N is a near-field and A is an N-sub near-field space in the near-field spaces sense, that is (A, +) is not necessarily abelian, then AN⊗ B and A ⊗NB can be constructed, which may be different.
The structure of tensor products of near-field spaces can be explored further and these proofs here to show the importance of this concept.
ACKNOWLEDGEMENT
The author being a Professor is indebted to the referee for his various valuable comments leading to the improvement of the advanced research article. This work was supported by the chairman Sri B Srinivasa Rao, Kakinada Institute of Technology & Science (K.I.T.S.), R&D education Department S&H (Mathematics), Divili 533 433. Andhra Pradesh INDIA.
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- Near Rings (IFPINR-delta-NR)”, Int. J. of Contemporary Mathematics, Copyright @ Mind Reader Publications, ISSN No: 0973-6298, Vol. 2, No. 1, pp.53-58, June 2011.18. N V Nagendram, B Ramesh paper "A Note on Asymptotic value of the Maximal size of a Graph with rainbow connection number 2*(AVM-SGR-CN2*)" published in an International Journal of Advances in Algebra(IJAA) Jordan @ Research India Publications, Rohini , New Delhi, ISSN 0973-6964 Volume 5, Number 2 (2012), pp. 103-112.
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23. N V Nagendram, Ch Padma, Dr T V Pradeep Kumar and Dr Y V Reddy "A Note on Pi-Regularity and Pi-S-Unitality over Noetherian Regular Delta Near Rings (PI-R-PI-S-U-NR-Delta-NR)" Published in International Electronic Journal of Pure and Applied Mathematics, IeJPAM-2012-17-669 ISSN-1314-0744 Vol-75 No-4 (2011).
24. N V Nagendram, Ch Padma, Dr T V Pradeep Kumar and Dr Y V Reddy "Ideal Comparability over Noetherian Regular Delta Near Rings(IC-NR-Delta-NR)" Published in International Journal of Advances in Algebra (IJAA, Jordan), ISSN 0973-6964 Vol:5,NO:1(2012), pp.43-53@Research India publications, Rohini, New Delhi.
25. N. V. Nagendram , S. Venu Madava Sarma and T. V. Pradeep Kumar, “A Note On Sufficient Condition Of Hamiltonian Path To Complete Graphs (SC-HPCG)”, IJMA-2(11), 2011, pp.1-6.
26. N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Noetherian Regular Delta Near Rings and their Extensions(NR-delta-NR)”, IJCMS, Bulgaria, IJCMS-5-8-2011,Vol.6,2011,No.6,255-262.
27. N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On Semi Noehterian Regular Matrix Delta Near Rings and their Extensions (SNRM-delta-NR)”,Jordan,@ResearchIndiaPublications,AdvancesinAlgebraISSN 0973-6964 Volume 4, Number 1 (2011), pp.51-55© Research India Publicationspp.51-55.
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31. N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy “On IFP Ideals on Noetherian Regular Delta Near Rings(IFPINR-delta-NR)”, Int. J. of Contemporary Mathematics, Vol. 2, No. 1-2, Jan-Dec 2011, Copyright @ Mind Reader Publications, ISSN No: 0973-6298, pp.43-46.
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Noetherian regular Delta-near rings (GIBDNR- d-NR)" , International Journal of Theoretical Mathematics and Applications (TMA)accepted and published by TMA, Greece, Athens,ISSN:1792- 9687 (print),vol.1, no.1, 2011, 59-71, 1792-9709 (online), International Scientific Press, 2011.
36. N V Nagendram, Dr T V Pradeep Kumar and Dr Y V Reddy "Inversive Localization of Noetherian regular Delta-near rings (ILNR- Delta-NR)" , International Journal of Pure And Applied Mathematics published by IJPAM-2012-17-668, ISSN.1314-0744 vol-75 No-3,SOFIA, Bulgaria.
37. N VNagendram1, N Chandra Sekhara Rao2 "Optical Near field Mapping of Plasmonic Nano Prisms over Noetherian Regular Delta Near Fields (ONFMPN-NR-Delta-NR)" accepted for 2nd international Conference by International Journal of Mathematical Sciences and Applications, IJMSA @ mind reader publications, New Delhi going to conduct on 15 – 16 th December 2012 also for publication.
38. N V Nagendram, K V S K Murthy (Yoga), "A Note on Present Trends on Yoga Apart From Medicine Usage and Its Applications (PTYAFMUIA)" Published by the International Association of Journal of Yoga Therapy, IAYT 18 th August, 2012.
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δ
-NR)" accepted for 3nd international Conference by International Journal of Mathematical Sciences and Applications, IJMSA@mind reader publications, New Delhi going to conduct on 15 – 16 th December 2014 also for publication.41. N V Nagendram “Characterization of near-field spaces over Baer-ideals" accepted for 4th international Conference by International Journal Conference of Mathematical Sciences and Applications, IJCMSA @ mind reader publications, New Delhi going to conduct on 19 – 20 th December 2015 at Asian Institute of Technology AIT, Klaung Lange 12120, Bangkok, Thailand.
42. N V Nagendram, S V M Sarma Dr T V Pradeep Kumar “A note on sufficient condition of Hamiltonian path to Complete Graphs” published in International Journal of Mathematical archive IJMA, ISSN 2229-5046, Vol.2, No.2, Pg. 2113 – 2118, 2011.
43. N V Nagendram, S V M Sarma, Dr T V Pradeep Kumar "A note on Relations between Barnette and Sparse Graphs" publishd in an International Journal of Mathematical Archive (IJMA), An International Peer Review Journal for Mathematical, Science & Computing Professionals, 2(12), 2011, pg no.2538-2542, ISSN 2229 – 5046.
44. N V Nagendram "On Semi Modules over Artinian Regular Delta Near Rings(S Modules-AR-Delta-NR) Accepted and published in an International Journal of Mathematical Archive (IJMA)", An International Peer Review Journal for Mathematical, Science & Computing Professionals ISSN 2229-5046, IJMA-3-474, 2012. 45. N V Nagendram "A note on Generating Near-field efficiently Theorem from Algebraic K - Theory" published
by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.3, No.10, Pg. 1 – 8, 2012. 46. N V Nagendram and B Ramesh on "Polynomials over Euclidean Domain in Noetherian Regular Delta Near
Ring Some Problems related to Near Fields of Mappings (PED-NR-Delta-NR & SPR-NF)" Accepted and published in an International Journal of Mathematical Archive (IJMA), An International Peer Review Journal for Mathematical, Science & Computing Professionals ISSN 2229-5046, vol.3, no.8, pp no. 2998-3002, 2012. 47. N V Nagendram "Semi Simple near-fields Generating efficiently Theorem from Algebraic K - Theory"
published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.3, No.12, Pg. 1 – 7, 2012.
48. N V Nagendram "---" published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.3, No.10, Pg. 3612 – 3619, 2012.
50. N VNagendram, Dr T V Pradeep Kumar, “Compactness in fuzzy near-field spaces (CN-F-NS)”, IJMA, ISSN. 2229 – 5046, Vol.4, No.10, Pg. 1 – 12, 2013.
51. N V Nagendram,Dr T V Pradeep Kumar and Dr Y Venkateswara Reddy, " Fuzzy Bi-
Γ
ideals inΓ
semi near – field spaces (F Bi-Gamma I G)" published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.4, No.11, Pg. 1 – 11, 2013.52. N V Nagendram," EIFP Near-fields extension of near-rings and regular delta near-rings (EIFP-NF-E-NR)“ published by International Journal of Mathematical Archive, IJMA, ISSN. 2229 - 5046, Vol.4, No.8, Pg. 1 –11, 2013.
53. N V Nagendram, E Sudeeshna Susila, “Generalization of (
∈
,∈
Vqk) fuzzy sub fields and ideals of near-fields(GF-NF-IO-NF)”, IJMA, ISSN.2229-5046, Vol.4, No.7,Pg. 1 – 11, 2013.54. N V Nagendram,Dr T V Pradeep Kumar," A note on Levitzki radical of near-fields(LR-NF)", Published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.4, No.4, Pg.288 – 295, 2013. 55. N V Nagendram, "Amalgamated duplications of some special near-fields (AD-SP-N-F)", Published by
International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.4, No.2, Pg.1 – 7, 2013.
56. N V Nagendram," Infinite sub near-fields of infinite near-fields and near-left almost near-fields(IS-NF-INF-NL-A-NF)",Published by International Journal of Mathematical Archive, IJMA,ISSN. 2229-5046, Vol.4, No.2, Pg. 90 – 99, 2013.
57. N V Nagendram "---" published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.4, No.7, Pg. 1 – 11, 2013
58. N V Nagendram, E Sudeeshna Susila, Dr T V Pradeep Kumar “Some problems and applications of ordinary differential equations to Hilbert Spaces in Engg mathematics (SP-O-DE-HS-EM)”, IJMA, ISSN.2229-5046, Vol.4, No.4,Pg. 118 – 125, 2013.
59. N V Nagendram, Dr T V Pradeep Kumar and D Venkateswarlu, " Completeness of field spaces over near-fields (VNFS-O-NF)" published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.5, No.2, Pg. 65 – 74, 2014
60. Dr N V Nagendram "A note on Divided near-field spaces and
φ
-pseudo – valuation near-field spaces over regularδ
-near-rings (DNF-φ
-PVNFS-O-δ
-NR)" published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.6, No.4, Pg. 31 – 38, 2015.61. Dr. N V Nagendram “A Note on B1-Near-fields over R-delta-NR (B1-NFS-R-
δ
-NR)", Published byInternational Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.6, No.8, Pg. 144 – 151, 2015. 62. Dr. N V Nagendram " A Note on TL-ideal of Near-fields over R-delta-NR (TL-I-NFS-R-
δ
-NR)", Published byInternational Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.6, No.8, Pg. 51 – 65, 2015. 63. Dr. N V Nagendram "A Note on structure of periodic Near-fields and near-field spaces(ANS-P-NF-NFS)",
Published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.7, No.4, Pg. 1 – 7, 2016.
64. Dr. N V Nagendram "Certain Near-field spaces are Near-fields(C-NFS-NF)", Published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.7, No.4, Pg. 1 – 7, 2016.
65. Dr. N V Nagendram "Sum of Annihilators Near-field spaces over Near-rings is Annihilator Near-field space (SA-NFS-O-A-NFS)", Published by International Journal of Mathematical Archive, IJMA, ISSN. 2229-5046, Vol.7, No.1, Pg. 125 – 136, 2016.
66. Dr. N V Nagendram " A note on commutativity of periodic near-field spaces", Published by IJMA, ISSN. 2229 - 5046, Vol.7, No. 6, Pg. 27 – 33, 2016.
67. Dr N V Nagendram “Densely Co-Hopfian sub near-field spaces over a near-field, IMA, ISSN No.2229-5046, 2016, Vol.7, No.10, Pg 1-12.
68. Dr N V Nagendram, “Closed (or open) sub field spaces of commutative field space over a near-field”, 2016, Vol.7, No, 9, ISSN NO. 2229 – 5046, Pg No.57 – 72.
69. Dr N V Nagendram, “Homomorphism of near-field spaces over a near-field “IJMA Jan 2017, Vol.8, No, 2, ISSN NO.2229 – 5046, Pg No. 141 – 146.
70. Dr N V Nagendram, “Sigma – toe derivations of near-field spaces over a near-field “IJMA Jan 2017, Vol.8, No, 4, ISSN NO.2229 – 5046, Pg No. 1 – 8.
71. Dr N V Nagendram, “On the hyper center of near-field spaces over a near-field “IJMA Feb 2017, Vol.8, No, 2, ISSN NO.2229 – 5046, Pg No. 113 – 119.
Source of support: Nil, Conflict of interest: None Declared.
